i)
Any solution is unambiguously labeled by an arbitrary function of \(p-2\)  \(T\) -variables. This function can be associated with the free energy at all \(t_k=0\) . We call it the bare free energy \({\cal F}^{(0)}(T)\) .
ii)
Solutions to the Virasoro constraints (or loop equations) are constructed from \({\cal F}^{(0)}(T)\)  by an evolution operator \(\hat{U}(T,t)\)  that does not depend on \({\cal F}^{(0)}(T)\) :
\(Z(T,t)=\hat{U}(T,t){\rm e}^{{\cal F}^{(0)}(T)}.\) 
iii)
The evolution operator \(\hat{U}(T,t)\)  is understood here as a power series in \(t_k\)  with the coefficients which can be completely expressed in terms of the unique operator \(\check{R} (x)\)  with its non-local “function” \(\check{y}\)
\(\check{y}\equiv \sqrt{W^{\prime }(x)^2 -4\check{R}(x)},\qquad \check{R} (x)\equiv -\sum _{a,b=0}(a+b+2)T_{a+b+2}x^a{\partial \over \partial T_b},\) 
its derivatives and \(W^{\prime }(x)\) . Here \(\check{y}\)  is defined as a power series at large \(x\) , see [1]}, [2]} for the details.

